Precoding schemes are widely used in multiple-input multiple-output (MIMO) wireless systems. The precoded data achieves higher transmission reliability when transmitted via wireless channel, due to the coding distance between the signals. The receiver is able to correct transmission errors when precoding is applied. Further, precoding in multi-antenna systems allow to send the data stream to multiple antennas concurrently.
Some codebook designs have been reported, which codebooks are designed based on, for example, Grassmannian manifold, vector quantization, Fourier transform and Kerdock manifold.
When we take it in a simple way, it could be said that the Grassmannian codebook design based on the Grassmannian manifold is substantially superior, because of a large distance between codewords. The Grassmannian codebook Wg(k) is optimal for selecting codewords wg(i) with a maximum codeword distance dw. Here we show one example of the Grassmannian codebook which is suitable for 2×2 wireless communication system; 2 antennas at a transmission side and 2 antennas at a receiver side.
                                          w            0            g                    =                      [                                                                                                      -                      0.1612                                        -                                          j                      ⁢                                                                                          ⁢                      0.7348                                                                                                                                  -                      0.5135                                        -                                          j                      ⁢                                                                                          ⁢                      0.4128                                                                                                                                                              -                      0.0787                                        -                                          j                      ⁢                                                                                          ⁢                      0.3192                                                                                                                                  -                      0.2506                                        +                                          j                      ⁢                                                                                          ⁢                      0.9106                                                                                            ]                          ⁢                                  ⁢                              w            1            g                    =                      [                                                                                                      -                      0.2399                                        +                                          j                      ⁢                                                                                          ⁢                      0.5985                                                                                                                                  -                      0.7641                                        -                                          j                      ⁢                                                                                          ⁢                      0.0212                                                                                                                                        -                    0.9541                                                                    0.2996                                                      ]                                              [                  Equation          ⁢                                          ⁢          1                ]            
Currently, the MIMO wireless communication system, for example, IEEE802.11, what has been referred to as Wi-Fi, is optimized for indoor usage. Indoor environments are typically characterized by significant multi-path propagation between sender and receiver due to reflections and attenuation by various obstacles, for example, walls, furniture, windows, mirrors, etc. Since the Grassmannian codebook Wg(k) is optimal for selecting codewords wg(i) with a maximum codeword distance dw, we can say that it is one of the best design choices.
However, a problem of the Grassmannian codebook lies in a significant large computational overhead because the codewords of Grassmannian manifold have a large number of digits. More specifically, a large number of digits generate a large number of multiplications, which requires a big overhead in signal processing and leads to additional processing time, thus increasing the latency in an entire communication system. As high speed and large capacity communication is now required, the Grassmannian codebook which requires longer processing time needs some alternative precoding scheme.
Kerdock codebook design which is based on Kerdock manifold therefore provides one prospective answer, because of its complexity reduction. Here we show one example of the Kerdock codebook which is suitable for a 2×2 wireless communication system.
                                          w            0            k                    =                                    1                              2                                      ⁡                          [                                                                    1                                                        1                                                                                        1                                                                              -                      1                                                                                  ]                                      ⁢                                  ⁢                              w            1            k                    =                                    1                              2                                      ⁡                          [                                                                    1                                                        1                                                                                        j                                                                              -                      j                                                                                  ]                                                          [                  Equation          ⁢                                          ⁢          2                ]            
Surely, taking the Kerdock codebook, each element of each codeword is fairly simple, which reduces storage and search requirements. In addition, according to NPTL 1 and PTL 1, system performance is approximately same between the Grassmannian codebook design and the Kerdock codebook design. It is because less channel state information (CSI) feedback is required for the Kerdock codebook, so it reduces the feedback overhead which otherwise would interfere with the data traffic. Therefore the Kerdock codebook leads to high performance comparable or better than previously known codebooks.